John R. Reay

The boundary characteristic and Pick's theorem in the Archimedean planar tilings

Abstract The tiling of the plane by unit squares is only one of the 11 Archimedean tilings which use regular polygons and have only one type of vertex. In this paper, the boundary characteristic of a lattice polygon is defined for every Archimedean tiling, and related enumeration formulae are found. Picks theorem (on the area of a lattice polygon in the tilling by squares) is then generalized for lattice polygons in each of the Archimedean tilings, by enumerating the number of tiles of each type in the polygon.

A Tverberg-type generalization of the Helly number of a convexity space

In 1966 H. Tverberg gave a far reaching generalization of the well-known classical theorem of J. Radon. In this paper a similar generalization of the classical Helly theorem is given and it is shown that among these two generalized theorems a relationship holds similar to a theorem proved by F.W. Levi in 1951. Also the generalized Helly theorem in the convex product and convex sum space are investigated.

Areas of GeneralizedH-Polygons

LetHbe the set of vertices of a tiling of the plane by regular hexagons of unit area. There is a Pick-type formula which can be applied only to a restricted class of polygons with corners inH. The main purpose of the paper is to give a Pick-type formula applicable to arbitraryH-polygons. The setting is that of general oriented polygons as introduced by Grunbaum and Shephard.

Hamiltonian Cycles in T-Graphs

Abstract. There is only one finite, 2-connected, linearly convex graph in the Archimedean triangular tiling that does not have a Hamiltonian cycle.

Primitive and mensurable hex-triangles

When the corners of a planar polygon P are restricted to lie in the set H of vertices of a tiling of the plane by hexagons of unit area, then very often the area of P is given by the Pick-type formula A(P)=b/4+i/2+c/12-1, where b and i count the number of points of H on the boundary ∂P and in the interior of P respectively, and c is the boundary characteristic. We now characterize all primitive triangles for which this formula holds, and consider the magnitude of the error when it fails.

Thin Hamiltonian cycles on Archimedean graphs

Abstract Archimedean graphs are finite subgraphs of an Archimedean tiling. If such a graph has a thin Hamiltonian cycle, then various area functions of the cycle are shown to be graph invariants, depending on the graph itself, but not on the particular thin Hamiltonian cycle.